I don't really know anything about pianos, but since I just got a digital one I guess I should give it a go.

The first thing I noticed is that it is difficult for me to press multiple keys that are a bit separated from each other with one hand. Not gonna lie: my hands are kinda small (I think).

I guess that it is still possible to play it well anyway and I just need practice. However, given that my hands are small, will my learning curve be significantly more challenging, or is this kind of problem very common and happens to most people?

The piano is absolutely easier to play with large hands (if your fingers aren't too broad) depending on how complex the piece is. Some pieces may be impossible to play without other tools if your hands are not large enough (notably Rachmaninoff).
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Matthew Read♦Dec 12 '12 at 0:37

3 Answers
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If your hands are really smaller than the "average hand" (whatever that is) there will be a few passages you'll find physically hard or impossible to play. These will be in the minority and probably feature in pieces of a more advanced complexity than you're going to meet when starting out.

Your feeling of "it is difficult for me to press multiple keys that are a bit separated from each other with one hand" is more likely just a lack of hand strength at this stage - whenever I try and teach a non-pianist a little piece I always forget that I only find it easy because my hands are used to making that sort of movement and shape.
When I don't play for a while and attempt something that requires a bit of dexterity, I find it a lot harder because the muscles have got lazy!

So don't worry too much about not being able to span 12 keys; work on finger-strength to compensate.

Of course it would help if I could remember his name, but I recently heard an interview with an internationally acclaimed concert pianist.

He said that because of his small hands, there were some passages where he struggled to reach the notes, so all of his practice went into learning to play the passage, rather than into developing the expression he wanted.

Yet his audience would love it anyway.

There will be many pieces for which your hand size will not be a problem. There will be some pieces that will require you to improve your stretch. There will be some pieces that will be unattainable for you.

After all, I could write a piece right now which no human could play with two hands!

In conclusion, there may be some parts you won't be able to play. But there will be plenty which you can play, so don't let it stop you learning the instrument.